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A300060 Number of domino tilings of the diagram of the integer partition with Heinz number n.

%I A300060 #21 May 28 2018 06:06:05
%S A300060 1,0,1,1,0,0,1,0,2,1,0,1,1,0,0,1,0,0,1,0,2,1,0,0,3,0,3,1,1,0,0,0,0,1,
%T A300060 0,2,1,0,2,1,0,0,1,0,0,1,0,1,5,0,0,1,1,0,3,0,2,0,0,0,1,1,3,1,0,0,0,0,
%U A300060 0,1,1,0,0,0,4,1,0,0,1,0,5,1,0,2,3,0,2,1,1,1,5,0,0,1,0,0,0,0,0,3,1,0,0,0,0,0
%N A300060 Number of domino tilings of the diagram of the integer partition with Heinz number n.
%C A300060 The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
%H A300060 Alois P. Heinz, <a href="/A300060/b300060.txt">Table of n, a(n) for n = 1..100000</a>
%H A300060 Wikipedia, <a href="https://en.wikipedia.org/wiki/Domino_tiling">Domino tiling</a>
%H A300060 Gus Wiseman, <a href="/A300060/a300060.png">The a(91) = 5 domino tilings of (6,4).</a>
%p A300060 h:= proc(l, f) option remember; local k; if min(l[])>0 then
%p A300060      `if`(nops(f)=0, 1, h(map(x-> x-1, l[1..f[1]]), subsop(1=[][], f)))
%p A300060     else for k from nops(l) while l[k]>0 by -1 do od;
%p A300060         `if`(nops(f)>0 and f[1]>=k, h(subsop(k=2, l), f), 0)+
%p A300060         `if`(k>1 and l[k-1]=0, h(subsop(k=1, k-1=1, l), f), 0)
%p A300060       fi
%p A300060     end:
%p A300060 g:= l-> `if`(add(`if`(l[i]::odd, (-1)^i, 0), i=1..nops(l))=0,
%p A300060         `if`(l=[], 1, h([0$l[1]], subsop(1=[][], l))), 0):
%p A300060 a:= n-> g(sort(map(i-> numtheory[pi](i[1])$i[2], ifactors(n)[2]), `>`)):
%p A300060 seq(a(n), n=1..120);  # _Alois P. Heinz_, May 22 2018
%t A300060 h[l_, f_] := h[l, f] = Module[{k}, If[Min[l] > 0, If[Length[f] == 0, 1, h[Map[Function[x, x-1], l[[Range @ f[[1]]]]], ReplacePart[f, 1 -> Nothing]]], For[k = Length[l], l[[k]] > 0, k-- ]; If[Length[f] > 0 && f[[1]] >= k, h[ReplacePart[l, k -> 2], f], 0] + If[k > 1 && l[[k-1]] == 0, h[ReplacePart[l, {k -> 1, k - 1 -> 1}], f], 0]]];
%t A300060 g[l_] := If[Sum[If[OddQ @ l[[i]], (-1)^i, 0], {i, 1, Length[l]}] == 0, If[l == {}, 1, h[Table[0, l[[1]]], ReplacePart[l, 1 -> Nothing]]], 0];
%t A300060 a[n_] := g[Reverse @ Sort[ Flatten[ Map[ Function[i, Table[PrimePi[i[[1]]], i[[2]]]], FactorInteger[n]]]]];
%t A300060 Array[a, 120] (* _Jean-François Alcover_, May 28 2018, after _Alois P. Heinz_ *)
%Y A300060 Cf. A000085, A000720, A000712, A000898, A001222, A004003, A056239, A099390, A138178, A153452, A238690, A296150, A296188, A299925, A299926, A300056, A300061, A304662.
%K A300060 nonn
%O A300060 1,9
%A A300060 _Gus Wiseman_, Feb 23 2018